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The cogrowth inequality from Whitehead's algorithm

Asif Shaikh

TL;DR

The paper investigates how cogrowth and entropy of a non-cyclic free factor $H\le F_m$ change under Ascari's refinement of Whitehead's algorithm. It models $H$ via the ergodic automaton $\mathcal{B}_H$ recognizing the language $L_H$ of reduced words and extends Ascari's edge-collapse technique to automata, producing $\mathcal{B}_{\varphi(H)}$ from $\mathcal{B}_H$ with an explicit matrix transformation. Using Perron–Frobenius theory, it proves a strict inequality $\lambda_H < \lambda_{\varphi(H)}$, equivalently $ent(L_H) < ent(L_{\varphi(H)})$ and $\alpha_H < \alpha_{\varphi(H)}$, showing that the automaton-cogrowth increases under Ascari-type refinements. The results provide a language-theoretic realization of core-graph reductions, include a detailed example, and raise open questions about maximizing cogrowth within automorphic orbits and extending the framework to broader classes of subgroups. This establishes a bridge between topological core-graph reductions and spectral properties of automata, with implications for the Whitehead maximal cogrowth problem and related dynamical aspects of free factors.

Abstract

This article focuses on free factors H <= F_m of the free group F_m with finite rank m > 2, and specifically addresses the implications of Ascari's refinement of the Whitehead automorphism phi for H as introduced in \cite{ascari2021fine}. Ascari showed that if the core Delta_H of H has more than one vertex, then the core Delta_{phi(H)} of phi(H) can be derived from Delta_H. We consider the regular language L_H of reduced words from F_m representing elements of H, and employ the construction of mathcal{B}_H described in \cite{DGS2021}. mathcal{B}_H is a finite ergodic, deterministic automaton that recognizes L_H. Extending Ascari's result, we show that for the aforementioned free factors H of F_m, the automaton mathcal{B}_{phi(H)} can be obtained from mathcal{B}_H. Further, we present a method for deriving the adjacency matrix of the transition graph of mathcal{B}_{phi(H)} from that of mathcal{B}_H and establish that alpha_H < alpha_{phi(H)}, where alpha_H, alpha_{phi(H)}$ represent the cogrowths of H and phi(H), respectively, with respect to a fixed basis X of F_m. The proof is based on the Perron-Frobenius theory for non-negative matrices.

The cogrowth inequality from Whitehead's algorithm

TL;DR

The paper investigates how cogrowth and entropy of a non-cyclic free factor change under Ascari's refinement of Whitehead's algorithm. It models via the ergodic automaton recognizing the language of reduced words and extends Ascari's edge-collapse technique to automata, producing from with an explicit matrix transformation. Using Perron–Frobenius theory, it proves a strict inequality , equivalently and , showing that the automaton-cogrowth increases under Ascari-type refinements. The results provide a language-theoretic realization of core-graph reductions, include a detailed example, and raise open questions about maximizing cogrowth within automorphic orbits and extending the framework to broader classes of subgroups. This establishes a bridge between topological core-graph reductions and spectral properties of automata, with implications for the Whitehead maximal cogrowth problem and related dynamical aspects of free factors.

Abstract

This article focuses on free factors H <= F_m of the free group F_m with finite rank m > 2, and specifically addresses the implications of Ascari's refinement of the Whitehead automorphism phi for H as introduced in \cite{ascari2021fine}. Ascari showed that if the core Delta_H of H has more than one vertex, then the core Delta_{phi(H)} of phi(H) can be derived from Delta_H. We consider the regular language L_H of reduced words from F_m representing elements of H, and employ the construction of mathcal{B}_H described in \cite{DGS2021}. mathcal{B}_H is a finite ergodic, deterministic automaton that recognizes L_H. Extending Ascari's result, we show that for the aforementioned free factors H of F_m, the automaton mathcal{B}_{phi(H)} can be obtained from mathcal{B}_H. Further, we present a method for deriving the adjacency matrix of the transition graph of mathcal{B}_{phi(H)} from that of mathcal{B}_H and establish that alpha_H < alpha_{phi(H)}, where alpha_H, alpha_{phi(H)}$ represent the cogrowths of H and phi(H), respectively, with respect to a fixed basis X of F_m. The proof is based on the Perron-Frobenius theory for non-negative matrices.
Paper Structure (17 sections, 16 theorems, 24 equations, 4 figures)

This paper contains 17 sections, 16 theorems, 24 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The automaton $\mathcal{B}_H$
  3. Preliminaries
  4. The Schireier and core graph of $H$
  5. Definition of $\mathcal{B}_H$
  6. Entropy of $L_H$
  7. Cogrowth of $H$
  8. Whitehead’s algorithm
  9. Whitehead automorphism
  10. Whitehead’s algorithm for free factors
  11. The Cogrowth Inequality
  12. An Extension of Ascari's Result
  13. The state enumeration
  14. The Inequality
  15. Example
  16. ...and 2 more sections

Key Result

Theorem A

Let $w$ be a cyclically reduced word, which is primitive and not a single letter. Then there is a Whitehead automorphism $\varphi$ such that the cyclic length of $\varphi(w)$ is strictly smaller than the cyclic length of $w$.

Figures (4)

  • Figure 1: The circled vertices in Figures \ref{['fig:coreH']} and \ref{['fig:corePhiH']} indicate the root vertex.
  • Figure 2: The Whitehead graph of $H$
  • Figure 3:
  • Figure 4:

Theorems & Definitions (28)

  • Theorem A: Whitehead
  • Theorem B
  • Theorem C
  • Theorem D
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.1
  • Theorem 3.2
  • ...and 18 more